arXiv research

In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric f…

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…

Weil-Petersson volumes vary continuously with weighted points on a projective line.

This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller spaces. Even though historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert ove…

A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Peter…

We extend Weil-Petersson theory to infinite type Teichmüller spaces.

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kähler moduli space. A few basic examples are studied. In particular, we identify …

In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case.

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichmüller space. We sketch a collection of results by other authors and then proceed to develop the properties of the WP CAT(0) geometry. Our approach includes a simplified proof of the Masur-Wolf …

The paper studies gradients of geodesic-length functions and systoles on Teichmüller spaces.

In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by…

A quick overview is provided on the current development of the WP metric geometry.

We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, …

The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.

On the thick part of the moduli space of Riemann surfaces, where there is a positive lower bound of the systole of the surface, we show that all Weil-Petersson Riemannian curvatures are bounded, independent of the genus of the surface.

A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…

We highlight recent progresses in the study of the Weil-Petersson (WP) geometry of finite dimensional Teichmüller spaces. For recent progress on and the understanding of infinite dimensional Teichmüller spaces the reader is directed to the recent work of Teo-Takhtajan. As part of the highlight, we also present possible…

We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem …

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…

A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the c…

We prove that every Weil-Petersson isometry of the Teichmuller space T(g,n) is induced by an element of the extended mapping class group; here 3g-3+n > 1 and (g,n) is not (1,2). Our method follows Ivanov's proof of the Royden's analogous theorem for the Teichmuller metric: we study the action of an isometry on the fron…

Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. …

We study the Kähler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently introduced. Deformations of simple branched coverings fit into t…

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting met…

We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-c…

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,α…

A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.

In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat Kähler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completi…

Random surfaces have a strong spectral gap with polynomial rate.

Study of large-$n$ asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.

The Teichmüller curve is the fiber space over Teichmüller space of closed Riemann surfaces, where the fiber over a point in Teichmüller space is the underlying surface. We derive formulas for sectional curvatures on the Teichmüller curve. In particular, our method can be applied to investigate the geometry of the Weil-…

This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.

The paper studies convex cocompact structures using Weil-Petersson flow.

We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmuller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments…

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a…

In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

Harmonic extension of Weil-Petersson circle homeomorphisms

Random hyperbolic surfaces have a spectral gap that approaches 1/4 as genus grows.

In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.

In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that t…

Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces o…

Study abelian varieties' Weil-Petersson metric asymptotics.

Harmonic mappings into Teichmuller spaces appear in the study of manifolds which are fibrations whose fibers are Riemann surfaces. In this article we will study the existence and uniquenesses questions of harmonic mappings into Teichmuller spaces, as well as some local and global behavior of the harmonic images induced…

The paper calculates curvature formulas for stable Higgs bundles.